Cant is a real word, and its meaning is nothing like the meaning of can’t. Don’t mistake the two or write one in place of the other. If you mean can not, you must use an apostrophe in the shortened form.
A compound sentence doesn’t have to be just two sentences joined by a conjunction. The second sentence is compound and made up of not two but three shorter sentences, so you need a comma to separate the first from the second and the second from the third.
The part beginning with it’s the app follows logically and smoothly from you’re woken up. There is no abrupt change in direction within the sentence, so an em-dash is incorrect. I used a semicolon, but a period would also work.
Yesterday you found a new app on your phone. It doesn’t open, you can’t delete it, and you don’t know what it is. Tonight, you’re woken by an alert on your phone; it’s the app notifying you of some very specific instructions.
This is still not quite right… Let’s see if (not that there’s any doubt!) it will work better once we tidy up the verb tenses.
Yesterday you found a new app on your phone. It didn’t open, you couldn’t delete it, and you didn’t know what it was. The next night, you were woken by an alert on your phone; it was the app notifying you of some very specific instructions.
If you want to eliminate very from your writing, replace very specific with precise.